Question: In $\triangle ABC$, we have $AC=BC=7$  and $AB=2$. Suppose that  $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$. What is  $BD$?
Solution: Let $\overline{CH}$  be an altitude of $\triangle ABC$. Applying the Pythagorean Theorem to $\triangle CHB$ and to $\triangle CHD$ produces \[
8^2 - (BD +1)^2 = CH^2 = 7^2 - 1^2 = 48, \quad \text{so} \quad (BD+1)^2 = 16.
\] Thus $BD = \boxed{3}$.

[asy]
unitsize(0.5cm);
pair A,B,C,D,H;
A=(0,0);
H=(1,0);
B=(2,0);
D=(5,0);
C=(1,6);
draw(A--C--D--cycle,linewidth(0.7));
draw(H--C--B--cycle,linewidth(0.7));
label("1",(0.5,0),N);
label("1",(1.5,0),N);
label("7",(0.5,3),NW);
label("7",(1.5,3),NE);
label("8",(3.5,3),NE);
label("$A$",A,S);
label("$B$",B,S);
label("$D$",D,S);
label("$H$",H,S);
label("$C$",C,N);
[/asy]